A377467
Number of perfect-powers x in the range 2^n < x < 2^(n+1).
Original entry on oeis.org
0, 0, 0, 1, 2, 2, 4, 6, 7, 10, 15, 23, 31, 41, 60, 81, 117, 165, 230, 321, 452, 634, 891, 1252, 1766, 2486, 3504, 4935, 6958, 9815, 13849, 19537, 27577, 38932, 54971, 77640, 109667, 154921, 218878, 309276, 437046, 617657, 872967, 1233895, 1744152, 2465546, 3485477
Offset: 0
The perfect-powers in each prescribed range (rows):
.
.
.
9
25 27
36 49
81 100 121 125
144 169 196 216 225 243
289 324 343 361 400 441 484
529 576 625 676 729 784 841 900 961 1000
The binary expansions for n >= 3 (columns):
1001 11001 100100 1010001 10010000 100100001
11011 110001 1100100 10101001 101000100
1111001 11000100 101010111
1111101 11011000 101101001
11100001 110010000
11110011 110111001
111100100
The version for squarefree numbers is
A077643.
The version for prime-powers is
A244508.
Including powers of 2 in the range gives
A377435.
The version for non-perfect-powers is
A377701.
The union of all numbers counted is
A377702.
A081676 gives the greatest perfect-power <= n.
A131605 lists perfect-powers that are not prime-powers.
A377468 gives the least perfect-power > n.
Cf.
A000015,
A013597,
A014210,
A014234,
A023055,
A031218,
A045542,
A052410,
A065514,
A069623,
A216765,
A246655,
A345531.
-
perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Table[Length[Select[Range[2^n+1,2^(n+1)-1],perpowQ]],{n,0,15}]
-
from sympy import mobius, integer_nthroot
def A377467(n):
def f(x): return int(1-sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
return f((1<Chai Wah Wu, Nov 05 2024
A377702
Perfect-powers except for powers of 2.
Original entry on oeis.org
9, 25, 27, 36, 49, 81, 100, 121, 125, 144, 169, 196, 216, 225, 243, 289, 324, 343, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1000, 1089, 1156, 1225, 1296, 1331, 1369, 1444, 1521, 1600, 1681, 1728, 1764, 1849, 1936, 2025, 2116, 2187, 2197
Offset: 1
The terms together with their prime indices begin:
9: {2,2}
25: {3,3}
27: {2,2,2}
36: {1,1,2,2}
49: {4,4}
81: {2,2,2,2}
100: {1,1,3,3}
121: {5,5}
125: {3,3,3}
144: {1,1,1,1,2,2}
169: {6,6}
196: {1,1,4,4}
216: {1,1,1,2,2,2}
225: {2,2,3,3}
243: {2,2,2,2,2}
289: {7,7}
324: {1,1,2,2,2,2}
A081676 gives the greatest perfect-power <= n.
A131605 lists perfect-powers that are not prime-powers.
A188951 counts perfect-powers less than 2^n.
A377468 gives the least perfect-power > n.
Cf.
A014210,
A014234,
A023055,
A045542,
A052410,
A065514,
A069623,
A246655,
A304521,
A336416,
A345531,
A366833.
-
Select[Range[1000],GCD@@FactorInteger[#][[All,2]]>1&&!IntegerQ[Log[2,#]]&]
-
from sympy import mobius, integer_nthroot
def A377702(n):
def bisection(f,kmin=0,kmax=1):
while f(kmax) > kmax: kmax <<= 1
while kmax-kmin > 1:
kmid = kmax+kmin>>1
if f(kmid) <= kmid:
kmax = kmid
else:
kmin = kmid
return kmax
def f(x): return int(n-2+x+(l:=x.bit_length())+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,l)))
return bisection(f,n+1,n+1) # Chai Wah Wu, Nov 06 2024
A377435
Number of perfect-powers x in the range 2^n <= x < 2^(n+1).
Original entry on oeis.org
1, 0, 1, 2, 3, 3, 5, 7, 8, 11, 16, 24, 32, 42, 61, 82, 118, 166, 231, 322, 453, 635, 892, 1253, 1767, 2487, 3505, 4936, 6959, 9816, 13850, 19538, 27578, 38933, 54972, 77641, 109668, 154922, 218879, 309277, 437047, 617658, 872968, 1233896, 1744153, 2465547, 3485478
Offset: 0
The perfect-powers in each prescribed range (rows):
1
.
4
8 9
16 25 27
32 36 49
64 81 100 121 125
128 144 169 196 216 225 243
256 289 324 343 361 400 441 484
512 529 576 625 676 729 784 841 900 961 1000
Their binary expansions (columns):
1 . 100 1000 10000 100000 1000000 10000000 100000000
1001 11001 100100 1010001 10010000 100100001
11011 110001 1100100 10101001 101000100
1111001 11000100 101010111
1111101 11011000 101101001
11100001 110010000
11110011 110111001
111100100
The union of all numbers counted is
A001597, without powers of two
A377702.
The version for squarefree numbers is
A077643.
These are the first differences of
A188951.
The version for prime-powers is
A244508.
Not counting powers of 2 gives
A377467.
The version for non-perfect-powers is
A377701.
A081676 gives the greatest perfect-power <= n.
A131605 lists perfect-powers that are not prime-powers.
A377468 gives the least perfect-power > n.
Cf.
A000015,
A013597,
A014210,
A014234,
A023055,
A031218,
A045542,
A052410,
A065514,
A069623,
A216765,
A345531,
A377434.
-
perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Table[Length[Select[Range[2^n,2^(n+1)-1],perpowQ]],{n,0,15}]
-
from sympy import mobius, integer_nthroot
def A377435(n):
if n==0: return 1
def f(x): return int(1-sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
return f((1<Chai Wah Wu, Nov 05 2024
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